Linear and nonlinear programming:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham [u.a.]
Springer
2016
|
| Edition: | 4. ed. |
| Series: | International series in operations research & management science
228 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XIII, 546 S. graph. Darst. 235 mm x 155 mm, 0 g |
| ISBN: | 3319188410 9783319188416 |
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Record in the Search Index
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| adam_text |
Titel: Linear and nonlinear programming
Autor: Luenberger, David G
Jahr: 2016
David G. Luenberger · Yinyu Ye
Linear and Nonlinear
Programming
Fourth Edition
ö Springer
Contents
1 Introduction 1
1.1 Optimization 1
1.2 Types of Problems 2
1.3 Size of Problems 5
1.4 Iterative Algorithms and Convergence 6
Part I Linear Programming
2 Basic Properties of Linear Programs 11
2.1 Introduction 11
2.2 Examples of Linear Programming Problems 14
2.3 Basic Solutions 19
2.4 The Fundamental Theorem of Linear Programming 20
2.5 Relations to Convexity 23
2.6 Exercises 27
3 The Simplex Method 33
3.1 Pivots 33
3.2 Adjacent Extreme Points 38
3.3 Determining a Minimum Feasible Solution 42
3.4 Computational Procedure: Simplex Method 45
3.5 Rinding a Basic Feasible Solution 49
3.6 Matrix Form of the Simplex Method 54
3.7 Simplex Method for Transportation Problems 56
3.8 Decomposition 68
3.9 Summary 72
3.10 Exercises 73
4 Duality and Complementarity 83
4.1 Dual Linear Programs 83
4.2 The Duality Theorem 86
ix
x Contents
4.3 Relations to the Simplex Procedura 88
4.4 Sensitivity and Complementary Slackness 92
4.5 Max Flow-Min Cut Theorem 94
4.6 The Dual Simplex Method 100
4.7 *The Primal-Dual Algorithm 102
4.8 Summary 106
4.9 Exercises 107
5 Inferior-Point Methods 115
5.1 Elements of Complexity Theory 117
5.2 *The Simplex Method Is Not Polynomial-Time 118
5.3 *The Ellipsoid Method 119
5.4 The Analytic Center 123
5.5 The Central Path 125
5.6 Solution Strategien 130
5.7 Termination and Initialization 137
5.8 Summary 142
5.9 Exercises 143
6 Conic Linear Programming 149
6.1 ConvexCones 149
6.2 Conic Linear Programming Problem 150
6.3 Parkas' Lemma for Conic Linear Programming 154
6.4 Conic Linear Programming Duality 158
6.5 Complementarity and Solution Rank of SDP 166
6.6 Interior-Point Algorithms for Conic Linear Programming 170
6.7 Summary 173
6.8 Exercises 174
Part II Unconstrained Problems
7 Basic Properties ofSolutions and Algorithms 179
7.1 First-Order Necessary Conditions 180
7.2 Examples of Unconstrained Problems 182
7.3 Second-Order Conditions 185
7.4 Convex and Concave Functions 188
7.5 Minimization and Maximization of Convex Functions 192
7.6 *Zero-Order Conditions 194
7.7 Global Convergence of Descent Algorithms 196
7.8 Speed of Convergence 204
7.9 Summary 209
7.10 Exercises 209
Contents xi
8 Basic Descent Methods 213
8.1 Line Search Algorithms 214
8.2 The Method of Steepest Descent 229
8.3 Applications of the Convergence Theory 239
8.4 Accelerated Steepest Descent 243
8.5 Newton's Method 245
8.6 Coordinate Descent Methods 252
8.7 Summary 257
8.8 Exercises 258
9 Conjugate Direction Methods 263
9.1 Conjugate Directions 263
9.2 Descent Properties of the Conjugate Direction Method 266
9.3 The Conjugate Gradient Method 268
9.4 The C-G Method as an Optimal Process 270
9.5 The Partial Conjugate Gradient Method 273
9.6 Extension to Nonquadratic Problems 276
9.7 *Parallel Tangents 279
9.8 Exercises 281
10 Quasi-Newton Methods 285
10.1 Modified Newton Method 286
10.2 Construction of the Inverse 288
10.3 Davidon-Fletcher-Powell Method 290
10.4 The Broyden Family 293
10.5 Convergence Properties 296
10.6 Scaling 300
10.7 Memoryless Quasi-Newton Methods 304
10.8 *Combination of Steepest Descent and Newton's Method 306
10.9 Summary 312
10.10 Exercises 313
Part III Constrained Minimization
11 Constrained Minimization Conditions 321
11.1 Constraints 321
11.2 Tangent Plane 323
11.3 First-Order Necessary Conditions (Equality Constraints) 326
11.4 Examples 327
11.5 Second-Order Conditions 333
11.6 Eigenvalues in Tangent Subspace 335
11.7 Sensitivity 338
11.8 Inequality Constraints 340
11.9 Zero-Order Conditions and Lagrangian Relaxation 344
11.10 Summary 351
11.11 Exercises 352
xii Contents
12 Primal Methods 357
12.1 Advantage of Primal Methods 357
12.2 Feasible Direction Methods 358
12.3 Active Set Methods 360
12.4 The Gradient Projection Method 364
12.5 Convergence Rate of the Gradient Projection Method 370
12.6 The Reduced Gradient Method 378
12.7 Convergence Rate of the Reduced Gradient Method 383
12.8 *Variations 390
12.9 Summary 392
12.10 Exercises 392
13 Penalty and Barrier Methods 397
13.1 Penalty Methods 398
13.2 Barrier Methods 401
13.3 Properties of Penalty and Barrier Functions 403
13.4 Newton's Method and Penalty Functions 412
13.5 Conjugate Gradients and Penalty Methods 413
13.6 Normalization of Penalty Functions 415
13.7 Penalty Functions and Gradient Projection 417
13.8 *Exact Penalty Functions 421
13.9 Summary 423
13.10 Exercises 425
14 Duality and Dual Methods 429
14.1 Global Duality 430
14.2 Local Duality 435
14.3 Canonical Convergence Rate of Dual Steepest Ascent 440
14.4 Separable Problems and Their Duals 441
14.5 Augmented Lagrangian 445
14.6 The Method of Multipliers 449
14.7 The Alternating Direction Method of Multipliers 454
14.8 *Cutting Plane Methods 458
14.9 Exercises 464
15 Primal-Dual Methods 467
15.1 The Standard Problem 467
15.2 A Simple Merit Function 470
15.3 Basic Primal-Dual Methods 471
15.4 Modified Newton Methods 477
15.5 Descent Properties 478
15.6 * Rate of Convergence 483
15.7 Primal-Dual Interior Point Methods 485
15.8 Summary 488
15.9 Exercises 489
Contents xiii
A Mathematical Review 495
A.l Sets 495
A.2 Matrix Notation 496
A.3 Spaces 497
A.4 Eigenvalues and Quadratic Forms 498
A.5 Topological Concepts 499
A.6 Functions 500
B Convex Sets 505
B.l Basic Definitions 505
B.2 Hyperplanes and Polytopes 507
B.3 Separating and Supporting Hyperplanes 509
B.4 Extreme Points 511
C Gaussian Elimination 513
D Basic Network Concepts 517
D.l Flows in Networks 519
D.2 Tree Procedure 519
D.3 Capacitated Networks 521
Bibliography 523
Index 539 |
| any_adam_object | 1 |
| author | Luenberger, David G. 1937- Ye, Yinyu |
| author_GND | (DE-588)135911338 (DE-588)170450880 |
| author_facet | Luenberger, David G. 1937- Ye, Yinyu |
| author_role | aut aut |
| author_sort | Luenberger, David G. 1937- |
| author_variant | d g l dg dgl y y yy |
| building | Verbundindex |
| bvnumber | BV042937459 |
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| ctrlnum | (OCoLC)927102532 (DE-599)DNB1069433403 |
| dewey-full | 519.72 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.72 |
| dewey-search | 519.72 |
| dewey-sort | 3519.72 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 4. ed. |
| format | Book |
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| id | DE-604.BV042937459 |
| illustrated | Illustrated |
| indexdate | 2024-08-03T02:43:58Z |
| institution | BVB |
| isbn | 3319188410 9783319188416 |
| language | English |
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| physical | XIII, 546 S. graph. Darst. 235 mm x 155 mm, 0 g |
| publishDate | 2016 |
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| publisher | Springer |
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| spelling | Luenberger, David G. 1937- Verfasser (DE-588)135911338 aut Linear and nonlinear programming David G. Luenberger ; Yinyu Ye 4. ed. Cham [u.a.] Springer 2016 XIII, 546 S. graph. Darst. 235 mm x 155 mm, 0 g txt rdacontent n rdamedia nc rdacarrier International series in operations research & management science 228 Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Nichtlineare Optimierung (DE-588)4128192-5 gnd rswk-swf Lineare Ordnung (DE-588)4167706-7 gnd rswk-swf Optimierung (DE-588)4043664-0 gnd rswk-swf Research ; Zielgruppe TBC ; BIC Subject Heading Semidefinite Programming Linear Programming Nonlinear Programming Mathematical Programming Operations Research Luenberger Optimization Models Optimierung (DE-588)4043664-0 s DE-604 Lineare Ordnung (DE-588)4167706-7 s Nichtlineare Optimierung (DE-588)4128192-5 s 1\p DE-604 Lineare Optimierung (DE-588)4035816-1 s 2\p DE-604 Ye, Yinyu Verfasser (DE-588)170450880 aut Erscheint auch als Online-Ausgabe 978-3-319-18842-3 International series in operations research & management science 228 (DE-604)BV011630976 228 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=5202445&prov=M&dok_var=1&dok_ext=htm Inhaltstext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028364056&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Luenberger, David G. 1937- Ye, Yinyu Linear and nonlinear programming International series in operations research & management science Lineare Optimierung (DE-588)4035816-1 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd Lineare Ordnung (DE-588)4167706-7 gnd Optimierung (DE-588)4043664-0 gnd |
| subject_GND | (DE-588)4035816-1 (DE-588)4128192-5 (DE-588)4167706-7 (DE-588)4043664-0 |
| title | Linear and nonlinear programming |
| title_auth | Linear and nonlinear programming |
| title_exact_search | Linear and nonlinear programming |
| title_full | Linear and nonlinear programming David G. Luenberger ; Yinyu Ye |
| title_fullStr | Linear and nonlinear programming David G. Luenberger ; Yinyu Ye |
| title_full_unstemmed | Linear and nonlinear programming David G. Luenberger ; Yinyu Ye |
| title_short | Linear and nonlinear programming |
| title_sort | linear and nonlinear programming |
| topic | Lineare Optimierung (DE-588)4035816-1 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd Lineare Ordnung (DE-588)4167706-7 gnd Optimierung (DE-588)4043664-0 gnd |
| topic_facet | Lineare Optimierung Nichtlineare Optimierung Lineare Ordnung Optimierung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=5202445&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028364056&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011630976 |
| work_keys_str_mv | AT luenbergerdavidg linearandnonlinearprogramming AT yeyinyu linearandnonlinearprogramming |